Invariants
Level: | $30$ | $\SL_2$-level: | $10$ | Newform level: | $900$ | ||
Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot10^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 10D1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.24.1.13 |
Level structure
$\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}17&0\\29&13\end{bmatrix}$, $\begin{bmatrix}26&25\\27&2\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 30-isogeny field degree: | $12$ |
Cyclic 30-torsion field degree: | $96$ |
Full 30-torsion field degree: | $5760$ |
Jacobian
Conductor: | $2^{2}\cdot3^{2}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 900.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + x z + 2 x w - y^{2} $ |
$=$ | $27 x^{2} - 4 x z - 8 x w + 10 y^{2} + 4 z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 35 x^{4} + 15 x^{3} y + 15 x^{2} y^{2} - 2 x^{2} z^{2} - 15 x y z^{2} + 4 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle y$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{55575716xz^{5}+1092428840xz^{4}w+2930693600xz^{3}w^{2}+1910000320xz^{2}w^{3}-65693120xzw^{4}-26277248xw^{5}-90233679z^{6}-289341588z^{5}w+171366300z^{4}w^{2}+917829280z^{3}w^{3}+449948400z^{2}w^{4}-10759488zw^{5}-3586496w^{6}}{7184xz^{5}+1454900xz^{4}w+4116305xz^{3}w^{2}+2059900xz^{2}w^{3}-1026455xzw^{4}-410582xw^{5}+259744z^{6}+739848z^{5}w+684420z^{4}w^{2}-166895z^{3}w^{3}-223545z^{2}w^{4}-168117zw^{5}-56039w^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
10.12.0.a.1 | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
15.12.0.b.2 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
30.12.1.f.1 | $30$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
30.72.1.j.2 | $30$ | $3$ | $3$ | $1$ | $1$ | dimension zero |
30.72.5.bd.2 | $30$ | $3$ | $3$ | $5$ | $1$ | $1^{2}\cdot2$ |
30.96.5.h.2 | $30$ | $4$ | $4$ | $5$ | $2$ | $1^{2}\cdot2$ |
30.120.5.o.1 | $30$ | $5$ | $5$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.96.5.cl.2 | $60$ | $4$ | $4$ | $5$ | $2$ | $1^{2}\cdot2$ |
150.120.5.f.1 | $150$ | $5$ | $5$ | $5$ | $?$ | not computed |
210.192.13.u.2 | $210$ | $8$ | $8$ | $13$ | $?$ | not computed |
330.288.21.u.1 | $330$ | $12$ | $12$ | $21$ | $?$ | not computed |